Computing with space vs space computing.
Space (real space, we all share) is not made of points. A point is an abstraction, the unattainable goal of a thought experiment, an atom of thought. Or a string of numbers (when we think with coordinates). Quantum physics tells us we can’t perform, even in principle, a physical experiment with the goal of exactly localizing the position of an object in space.
That’s pop philosophy. It might even be wrong (for example what quantum physics tells us is that we can’t perform physical experiments for localizing a particle in the phase space (position, momentum), not in the real space, whatever that means.
That’s also the turf of theoretical physicists, there are several, with various degree of mathematical soundness, theories about the structure of space. I shall not go in this direction, further.
Instead, I want to make a case for a biology inspired point of view. I made it before, repeatedly, starting with More than discrete or continuous: a bird’s view, but now I have a bit more tools to tackle it, and a bit of more patience to not hurry to conclusions.
So, if you prefer the red pill, then read this. Can you think about space in terms of what it does, not what it is? Can you describe space as seen by a fly, or by a toddler, or you need to stick to cartesian conventions and then fall into the trap of continuous vs discrete, and so on?
Think like this: you are a fly and you have 10^5 neurons and 10^7 synapses. You are very good at flying by using about 10-20 actuators, and you see really well because the most part of your brain is busy with that. Now, where in that brain and how exactly there is place for a representation of an euclidean 3d space? Mind you that humans have very little idea about how flies brains are capable of doing this and also, with their huge brains and their fast computers (much more faster and much bigger than a fly’s brain) were not successful yet to make a robot with the same competences as a fly. (They will make one, there is no magic involved, but the constraints are really hard: an autonomous fly which can survive with the energy consumption comparable with the one of a real fly, without computing or human exterior help, in a natural environment, for 24hrs, find food, avoid traps and eventually mate.)
So, after this motivating example, I state my hypothesis: whatever space is (and that’s a very hard and old problem), let’s not think about it passively (like a robot fly which is driven by some algorithms which use advanced human knowledge about euclidean geometry, systems of coordinates and the laws of mechanics) as being a receptacle, a something. Let’s think about space as described by what you can do in it.
The fly, for example, cannot possibly have a passive representation of space (and for us is the same) in the brain, but it does have the possibility to manipulate it’s actuators as a function of what it sees (i.e. of what it’s receptors perceive and send further to the brain) and of the state of it’s brain (and maybe on the history of that state, i.e. memory, stored in a mysterious way in the same tiny brain). However, actuators, sensors, brain and the environment are just one system, there is no ghost in, or outside that fly machine.
My hypothesis is that for the fly, that’s space. For us is the same, but we are far more complex than the fly. However, deep in our brains there are “patterns” (are they assemblies of neurons, are they patterns of synaptic activity, is it chemical, electric, …?) which are very basic (a child learns to see in the first months) and which are space, for us.
Now I’ll get mathematical. There are spaces everywhere in math, for example when we say: that’s a vector space, that’s a manifold, or even this is a group, a category, and so on. We say like this, but what we actually have (in the mind) is not a manifold, or a vector space, a group or category, but some short collections of patterns (rules, moves, axioms) which can be applied to the said objects. And that is enough for doing mathematics. This can be formalized, for example it’s enough to have some simple rules involving gates with two inputs and an output (the dilations) and we can prove that these simple rules describe all the concepts associated to any vector space, for example. and moreover not using at any moment any external knowledge. A dilation is simply the pattern of activities related to map making.
So, according to my hypothesis, a generic vector space is this collection of rules. When it comes to the dimension of it, there are supplementary relations to be added for being able to say that we speak about a 3d vector space, but it will be always about a generic 3d vector space. There is no concrete 3d space, when we say, for example, that we live in a 3d space, what we really say is that some of the things we can do in a generic 3d space can also be done in reality (i.e. we can perform experiments showing this, although there are again studies showing that our brain is almost as bad as concerns perceiving relations in the real space which correspond to theorems in geometry, as it is when we force it to do logic reasonings).
Conclusion for this part: there may be or not a passive space, the important thing is that when we think with space and about space, what we really do is using a collection of primitive patterns of thought about it.
Now, going to the Turing machine, it lacks space. Space can be used for enumeration, for example, but the Turing machine avoids this by supposing there is a tape (ordered set). It is proved that enumeration (i.e. the thing which resembles the most with space in the world of Turing machines) does not matter in the sense that it can be arbitrarily changed and still the conclusions and definitions from the field do not change. This is alike saying that the Turing machine is a geometrical object. But there is no geometrical description of a Turing machine (as far as I know) which is not using enumeration. This is alike saying that CS people can understand the concept of a sphere in terms of atlases , parametrizations and changes between those, but they can’t define spheres without them. Geometers can, and for this reason they can speak about what is intrinsically geometric about the sphere and what is only an artifact of the chosen coordinate system. In this sense, geometers are like flies: they know what can be done on a sphere without needing to know anything about coordinate systems.